Update docs for NeatContext 0.2.0 (context organizer, no in-app model)

[tanglearncode]

Why

NeatContext 0.2.0 hosts no model and no chat of its own. It organizes a team's context and serves it to the AI client the user already works in — Claude Code, Claude Desktop, Codex CLI, or ChatGPT Desktop — over a local MCP connection. That client reads the profiles, searches the knowledge folders, calls the extension tools, and writes the answer with its own model.

The published docs still described the removed architecture end-to-end: bring your own OpenAI-compatible model + API key, chat in tabs, streaming in-app answers. This PR brings every page in line with the shipped product (verified against apps/desktop and neatcontext_design_doc.md).

What changed

Core narrative

• intro, getting-started, core-concepts — reframed around Contexts + Library + connecting an external AI client; the "connect your model" step is gone (there is no model config anywhere in the app).

Features

• Replaced chats-and-tabs.md → contexts.md (a Context is the unit of handoff, shown as a tab).
• Replaced model-provider.md → connect-ai-clients.md (Claude Code / Claude Desktop / Codex CLI / ChatGPT Desktop, get_context, live edits, runtime boundary).
• Added library.md (personal + read-only Team Library, extension install lifecycle) and context-activity.md (per-Context activity log + retention).
• Updated domain-profiles, knowledge-bases, using-extensions, account for the Library/Context model and per-call, keychain-injected extension tool proxying.

Guides & extension authoring

• guides/incident-analysis — two Contexts + a connected AI client instead of two in-app chat tabs.
• extensions/* and the PagerDuty/Datadog guides — "the model in chat" / "top bar" / "Connect button in the chat" → connected AI client, Context selection, and activity log. connection_required now correctly describes the client relaying a "connect it first" prompt.
• Category descriptions updated.

Verification

• npm run build passes locally (Docusaurus onBrokenLinks: throw → no broken links; no MDX/admonition errors; all admonition titles use the required bracket syntax).

Follow-up (not in this PR)

• Screenshots: images of removed UI (chat tabs, model-provider page, in-app streamed answers, demo screenshots 03–08) are no longer referenced. Recapturing the Contexts page and the AI-client connection cards against a 0.2.0 build is a follow-up — no fabricated screenshots were added.